cp's OEIS Frontend

A008793 The problem of the calissons: number of ways to tile a hexagon of edge n with diamonds of side 1. Also number of plane partitions whose Young diagrams fit inside an n X n X n box.

%I A008793 #137 May 25 2025 19:24:05
%S A008793 1,2,20,980,232848,267227532,1478619421136,39405996318420160,
%T A008793 5055160684040254910720,3120344782196754906063540800,
%U A008793 9265037718181937012241727284450000,132307448895406086706107959899799334375000
%N A008793 The problem of the calissons: number of ways to tile a hexagon of edge n with diamonds of side 1. Also number of plane partitions whose Young diagrams fit inside an n X n X n box.
%C A008793 The 3-dimensional analog of A000984. - _William Entriken_, Aug 06 2013
%C A008793 The largest prime factor of a(n) is the largest prime p < 3*n. Its multiplicity is equal to 3*n-p. This can be proved with the formula of Michel Marcus, for example. - _Walter Trump_, Feb 11 2023
%C A008793 a(n) is also the number of resonance structures of circumcircum...coronene, where circum is repeated n-2 times where a(1) is the number of resonance structures of benzene (see Gutman et al.). - _Yuan Yao_, Oct 29 2023
%D A008793 Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 545, also p. 575 line -1 with a=b=c=n.
%D A008793 D. M. Bressoud, Proofs and Confirmations, Camb. Univ. Press, 1999; Eq. (6.8), p. 198. The first printing of Eq. (6.8) is wrong (see A049505 and A005157), but if one changes the limits in the formula (before it is corrected) to {1 <= i <= r, 1 <= j <= r}, one obtains the present sequence. - _N. J. A. Sloane_, Jun 30 2013
%D A008793 Gordon G. Cash and Jerry Ray Dias, Computation, Properties and Resonance Topology of Benzenoid Monoradicals and Polyradicals and the Eigenvectors Belonging to Their Zero Eigenvalues, J. Math. Chem., 30 (2001), 429-444. [See K, p. 442.]
%D A008793 Sebastien Desreux, Martin Matamala, Ivan Rapaport, Eric Remila, Domino tilings and related models: space of configurations of domains with holes, arXiv:math/0302344, 27 Feb 2003
%D A008793 Anne S. Meeussen, Erdal C. Oguz, Yair Shokef, Martin van Hecke1, Topological defects produce exotic mechanics in complex metamaterials, arXiv preprint 1903.07919, 2019 [See Section "Compatible metamaterials with fully antiferromagnetic interactions" - _N. J. A. Sloane_, Mar 23 2019]
%D A008793 J. Propp, Enumeration of matchings: problems and progress, pp. 255-291 in L. J. Billera et al., eds, New Perspectives in Algebraic Combinatorics, Cambridge, 1999 (see p. 261).
%H A008793 Seiichi Manyama, <a href="/A008793/b008793.txt">Table of n, a(n) for n = 0..54</a> (terms 0..30 from T. D. Noe)
%H A008793 T. Amdeberhan and V. H. Moll, <a href="https://doi.org/10.37236/1997">Arithmetic properties of plane partitions</a>, El. J. Comb. 18 (2) (2011) # P1.
%H A008793 Guy David and Carlos Tomei, <a href="https://www.jstor.org/stable/2325150">The Problem of the Calissons</a>, The American Mathematical Monthly, Vol. 96, No. 5 (May, 1989), pp. 429-431 (3 pages).
%H A008793 P. Di Francesco, P. Zinn-Justin and J.-B. Zuber, <a href="https://arxiv.org/abs/math-ph/0410002">Determinant Formulae for some Tiling Problems and Application to Fully Packed Loops</a>, arXiv:math-ph/0410002, 2004.
%H A008793 I. Fischer, <a href="https://arxiv.org/abs/math/9906102">Enumeration of rhombus tilings of a hexagon which contain a fixed rhombus in the center</a>, arXiv:math/9906102 [math.CO], 1999.
%H A008793 P. J. Forrester and A. Gamburd, <a href="https://arxiv.org/abs/math/0503002">Counting formulas associated with some random matrix averages</a>, arXiv:math/0503002 [math.CO], 2005.
%H A008793 M. Fulmek and C. Krattenthaler, <a href="https://arxiv.org/abs/math/9909038">The number of rhombus tilings of a symmetric hexagon which contain a fixed rhombus on the symmetry axis, II</a>, arXiv:math/9909038 [math.CO], 1999.
%H A008793 I. Gutman, S. J. Cyvin, and V. Ivanov-Petrovic, <a href="https://doi.org/10.1515/zna-1998-0810">Topological properties of circumcoronenes</a>, Z. Naturforsch., 53a, 1998, 699-703 (see p. 700) - _Emeric Deutsch_, May 14 2018
%H A008793 H. Helfgott and I. M. Gessel, <a href="https://arxiv.org/abs/math/9810143">Enumeration of tilings of diamonds and hexagons with defects</a>, arXiv:math/9810143 [math.CO], 1998.
%H A008793 Sam Hopkins and Tri Lai, <a href="https://arxiv.org/abs/2007.05381">Plane partitions of shifted double staircase shape</a>, arXiv:2007.05381 [math.CO], 2020. See Table 1 p. 9.
%H A008793 C. Krattenthaler, <a href="https://arxiv.org/abs/math/0503507">Advanced Determinant Calculus: A Complement</a>, Linear Algebra Appl. 411 (2005), 68-166; arXiv:math/0503507v2 [math.CO], 2005.
%H A008793 P. A. MacMahon, <a href="http://www.archive.org/details/combinatoryanaly02macmuoft">Combinatory Analysis, vol. 2</a>, Cambridge University Press, 1916; reprinted by Chelsea, New York, 1960.
%H A008793 Anne S. Meeussen, Erdal C. Oguz, Yair Shokef, and Martin van Hecke, <a href="https://arxiv.org/abs/1903.07919">Topological defects produce exotic mechanics in complex metamaterials</a>, arXiv:1903.07919 [cond-mat.soft], 2019.
%H A008793 James Propp, Enumeration of matchings: problems and progress, in L. J. Billera et al. (eds.), <a href="http://www.msri.org/publications/books/Book38/contents.html">New Perspectives in Algebraic Combinatorics</a>
%H A008793 James Propp, <a href="http://faculty.uml.edu/jpropp/update.pdf">Updated article</a>
%H A008793 James Propp, <a href="https://faculty.uml.edu//jpropp/rutgers22.pdf">Tiling Problems, Old and New</a>, Rutgers University Math Colloquium, March 30, 2022
%H A008793 N. C. Saldanha and C. Tomei, <a href="https://arxiv.org/abs/math/9801111">An overview of domino and lozenge tilings</a>, arXiv:math/9801111 [math.CO], 1998.
%H A008793 P. J. Taylor, <a href="http://cheddarmonk.org/papers/distinct-dimer-hex-tilings.pdf">Counting distinct dimer hex tilings</a>, Preprint, 2015.
%H A008793 Walter Trump, <a href="/A008793/a008793.txt">Prime factorization of a(n) for 1..950</a>
%H A008793 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PlanePartition.html">Plane Partition.</a>
%F A008793 a(n) = Product_{i = 0..n-1} (i^(-i)*(n+i)^(2*i-n)*(2*n+i)^(n-i)).
%F A008793 a(n) = Product_{i = 1..n} Product_{j = 0..n-1} (3*n-i-j)/(2*n-i-j).
%F A008793 a(n) = Product_{i = 1..n} Gamma[i]*Gamma[i+2*n]/Gamma[i+n]^2.
%F A008793 a(n) = Product_{i = 0..n-1} i!*(i+2*n)!/(i+n)!^2.
%F A008793 a(n) = Product_{i = 1..n} Product_{j = n..2*n-1} i+j / Product_{j = 0..n-1} i+j. - _Paul Barry_, Jun 13 2006
%F A008793 For n >= 1, a(n) = det(binomial(2*n,n+i-j)) for 1<=i,j<=n [Krattenhaller, Theorem 4, with a = b = c = n].
%F A008793 Let H(n) = Product_{k = 1..n-1} k!. Then for a,b,c nonnegative integers (H(a)*H(b)*H(c)*H(a+b+c))/(H(a+b)*H(b+c)*H(c+a)) is an integer [MacMahon, Chapter II, Section 429, p. 182, with x -> 1]. Setting a = b = c = n gives the entries for this sequence. - _Peter Bala_, Dec 22 2011
%F A008793 a(n) ~ exp(1/12) * 3^(9*n^2/2 - 1/12) / (A * n^(1/12) * 2^(6*n^2 - 1/4)), where A = A074962 = 1.28242712910062263687534256886979... is the Glaisher-Kinkelin constant. - _Vaclav Kotesovec_, Feb 27 2015
%F A008793 a(n) = Product_{i = 1..n} Product_{j = 1..n} (n+i+j-1)/(i+j-1). - _Michel Marcus_, Jul 13 2020
%F A008793 Conjecture: the supercongruences a(n*p^r) == a(n*p^(r-1))^p (mod p^(4*r)) hold for all primes p and positive integers n and r. - _Peter Bala_, Apr 07 2022
%p A008793 A008793 := proc(n) local i; mul((i - 1)!*(i + 2*n - 1)!/((i + n - 1)!)^2, i = 1 .. n) end proc;
%t A008793 Table[ Product[ (i+j+k-1)/(i+j+k-2), {i, n}, {j, n}, {k, n} ], {n, 10} ]
%o A008793 (PARI) a(n) = prod(i=1,n, prod(j=1, n, (n+i+j-1)/(i+j-1))); \\ _Michel Marcus_, Jul 13 2020
%Y A008793 Cf. A000984, A066931, A352656, A352657. Main diagonal of array A103905.
%K A008793 nonn,easy,nice
%O A008793 0,2
%A A008793 _Jonas Wallgren_
%E A008793 More terms from _Eric W. Weisstein_